The queueing theory for medical examinations.

The queueing theory for medical examinations.
(Copyright (C) Hee Cheong Lee All Rights Reserved.)

I have considered a queuing system model as follows :

A patient arrives and waits for a medical examination service given by one doctor.
After a medical examination service, the patient exits the queuing model.

Let x be the number of patients' arrival per an hour and let y be a mean time for a medical examination service (a minute) per a patient.
Then, you can express an average arrival interval as follows:

an average arrival interval = 60 / x (minute)

Therefore, in this case service using rate is expressed as follows:

a service using rate = average service time / average arrival interval

= (x * y) / 60.

Then, the average number of waiting patients is expressed as follows:

the average number of waiting patients = service using rate / (1 - service using rate)

= ((x * y) / 60) / (1 - ((x * y) / 60))

= (x * y) / (60 - (x * y))

( 0 < service using rate < 1 )

That means, an average waiting time per a patient is "(x * y * y) / (60 - (x * y))".

( 0 < x * y < 60 )

Well, let's think about how we get an average waiting time per a patient within 1 minute.

an average waiting time per a patient = (x * y * y) / (60 - (x * y))

<-> (x * y * y) / (60 - (x * y)) = 1

<-> (x * y * y) = (60 - (x * y))

<-> (x * y * y) + (x * y) - 60 = 0

This is a root of the function f(y) using the quadratic formula :
y = (root((x * x) + (240 * x)) - x) / (2 * x)

If x is equal to 10, then y is equal to 2, and this means 10 patients arrive per an hour, and
we should examine within 2 minutes in order to get patient's waiting time within 1 minute.

Let z be an average waiting time per a patient so as to generalize this theory.

Let us suppose you want to have an average waiting time per a patient within "z" minutes.
Then, the function mentioned above becomes as follows :

(x * y * y) / (60 - (x * y)) = z

<-> (x * y * y) = ((60 - (x * y)) * z)

<-> (x * y * y) + ((x * z) * y) - (60 * z) = 0

<-> y = (root((x * x) + (240 * x * z)) - (x * z)) / (2 * x)

For example:

Suppose you want to have an average waiting time per a patient within 2 minutes and 20 patients are expected to come per one hour.
You should examine a patient within 1.5 minutes because

y = (root((20 * 20) + (240 * 20 * 2)) - (20 * 2)) / (2 * 20)
<-> y = 1.5 (minutes)

<SUMMARY>

x : the number of patients' arrival per an hour
y : a mean time for a medical examination service (a minute) per a patient
z : an average waiting time per a patient

y = (root((x * x) + (240 * x * z)) - (x * z)) / (2 * x)

I have named this formula "THE FORMULA OF MEDICAL EXAMINATION SERVICE TIME". :)